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Just Few States are Enough: Randomized Sparse Feedback for Stability of Dynamical Systems

Zaid Hadach, Hajar El Hammouti, El Houcine Bergou, Adnane Saoud

TL;DR

This work addresses stabilization of linear discrete-time systems when the feedback controller only accesses a randomly selected subset of the state at each time step. It develops an $LMI$-based framework to jointly design a stabilizing gain $K$ and a sparsification policy, either uniform or adaptive, and proves sufficient conditions for asymptotic mean-square stability ($AMSS$) through the quantities $f(p)$ and $g(p_1,...,p_n)$. The authors propose two offline algorithms to compute the minimal sparseness level $p^\star$ and the per-component sparsification vector, enabling AMSS with dramatically reduced sensor usage (as low as $0.3\%$ in some simulations). Numerical results on grid-forming converters and a large-scale power system demonstrate that randomized sparse feedback can achieve performance comparable to full-state feedback while reducing measurement needs by several orders of magnitude, highlighting practical gains for large-scale dynamical networks. The framework extends to adaptive sparsification, offering component-wise prioritization based on control influence, and sets the stage for robust extensions to noisy or more complex objectives.

Abstract

While classical control theory assumes that the controller has access to measurements of the entire state (or output) at every time instant, this paper investigates a setting where the feedback controller can only access a randomly selected subset of the state vector at each time step. Due to the random sparsification that selects only a subset of the state components at each step, we analyze the stability of the closed-loop system in terms of Asymptotic Mean-Square Stability (AMSS), which ensures that the system state converges to zero in the mean-square sense. We consider the problem of designing both a feedback gain matrix and a measurement sparsification strategy that minimizes the number of state components required for feedback, while ensuring AMSS of the closed-loop system. Interestingly, (1) we provide conditions on the dynamics of the system under which it is possible to find a sparsification strategy, and (2) we propose a Linear Matrix Inequality (LMI) based algorithm that jointly computes a stabilizing gain matrix, and a randomized sparsification strategy that minimizes the expected number of measured state coordinates while preserving the AMSS. Our approach is then extended to the case where the sparsification probabilities vary across the state components. Based on these theoretical findings, we propose an algorithmic procedure to compute the vector of sparsification parameters, along with the corresponding feedback gain matrix. To the best of our knowledge, this is the first study to investigate the stability properties of control systems that rely solely on randomly selected state measurements. Numerical simulations demonstrate that, in some settings, the system achieves comparable performance to full-state feedback while requiring measurements from only $0.3\%$ of the state coordinates.

Just Few States are Enough: Randomized Sparse Feedback for Stability of Dynamical Systems

TL;DR

This work addresses stabilization of linear discrete-time systems when the feedback controller only accesses a randomly selected subset of the state at each time step. It develops an -based framework to jointly design a stabilizing gain and a sparsification policy, either uniform or adaptive, and proves sufficient conditions for asymptotic mean-square stability () through the quantities and . The authors propose two offline algorithms to compute the minimal sparseness level and the per-component sparsification vector, enabling AMSS with dramatically reduced sensor usage (as low as in some simulations). Numerical results on grid-forming converters and a large-scale power system demonstrate that randomized sparse feedback can achieve performance comparable to full-state feedback while reducing measurement needs by several orders of magnitude, highlighting practical gains for large-scale dynamical networks. The framework extends to adaptive sparsification, offering component-wise prioritization based on control influence, and sets the stage for robust extensions to noisy or more complex objectives.

Abstract

While classical control theory assumes that the controller has access to measurements of the entire state (or output) at every time instant, this paper investigates a setting where the feedback controller can only access a randomly selected subset of the state vector at each time step. Due to the random sparsification that selects only a subset of the state components at each step, we analyze the stability of the closed-loop system in terms of Asymptotic Mean-Square Stability (AMSS), which ensures that the system state converges to zero in the mean-square sense. We consider the problem of designing both a feedback gain matrix and a measurement sparsification strategy that minimizes the number of state components required for feedback, while ensuring AMSS of the closed-loop system. Interestingly, (1) we provide conditions on the dynamics of the system under which it is possible to find a sparsification strategy, and (2) we propose a Linear Matrix Inequality (LMI) based algorithm that jointly computes a stabilizing gain matrix, and a randomized sparsification strategy that minimizes the expected number of measured state coordinates while preserving the AMSS. Our approach is then extended to the case where the sparsification probabilities vary across the state components. Based on these theoretical findings, we propose an algorithmic procedure to compute the vector of sparsification parameters, along with the corresponding feedback gain matrix. To the best of our knowledge, this is the first study to investigate the stability properties of control systems that rely solely on randomly selected state measurements. Numerical simulations demonstrate that, in some settings, the system achieves comparable performance to full-state feedback while requiring measurements from only of the state coordinates.

Paper Structure

This paper contains 24 sections, 5 theorems, 40 equations, 12 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related work
  3. Preliminaries
  4. Mathematical Notations
  5. Randomized Sparsification Strategies
  6. Asymptotic Stability of the Expected State Vector
  7. Problem Formulation
  8. Main Results
  9. Adaptive sparsification
  10. Numerical Results
  11. Grid-Forming Converter
  12. Large-scale power system
  13. Conclusion
  14. Appendix A
  15. Proof of Lemma \ref{['expected_stability']}
  16. ...and 9 more sections

Key Result

Lemma 1

Consider the discrete-time control system $\Sigma$ in Eq.eqn:sys under the controller $k\mapsto u(k)=K\mathcal{C}(k)x(k)$, for a gain $K \subseteq \mathbb{R}^{n \times m}$ and where the compressor $k \mapsto \mathcal{C}(k)$ is defined by $\mathcal{C}(k)= Diag\left(\frac{c_1(k)}{p_1}, \ldots,\frac{c_ Moreover, if $K \in \mathcal{K}(A,B)$, then the system is asymptotically stable in expectation.

Figures (12)

  • Figure 1: Time evolution of the expected state components. The figure depicts stability for the case where $p = p^{\star} = 0.79$
  • Figure 2: Time evolution of the expected state components. The Figure Shows the stability of every component of the expected state under the adaptive sparsification strategy.
  • Figure 3: Variation of ${\mathbb{E}}_{x(0),{\mathcal{C}}}\left(\left\| x(k) \right\|_2^2\right)$ displayed for different values of the sparsification parameter.
  • Figure 4: Time evolution of the expected state components. It depicts stability for the case where $p = p^{\star} = 0.0026$.
  • Figure 5: Stability of components $\{1,400,900,1200,1600,2000\}$ of the expected state under the adaptive sparsification strategy.
  • ...and 7 more figures

Theorems & Definitions (10)

  • Lemma 1
  • Definition 1
  • Definition 2
  • Remark 1
  • Proposition 2
  • Theorem 1
  • Remark 2
  • Proposition 3
  • Theorem 2
  • Remark 3: Link to observability